Comparison of two different types of pseudomonotone mappings

The purpose of this paper is to show connections between two different types of pseudomonotone mappings. The first type of pseudomonotonicity notion, called in this paper A-pseudomonotonicity, was introduced by S. Karamardian [8] and has been frequently used in optimization problems [3, 4, 6, 11]. The second type, called in this paper B-pseudomonotonicity, is a generalization of the pseudomonotonicity notion introduced by H. Brézis [2] and used in the study of integral equations and partial differential equations [7, 9, 10]. We will follow some of the methods elaborated in [5] during the study of functional variational inequalities. Throughout this paper X will be a Banach space, X∗ its dual space, and K ⊂ X a nonempty, closed, convex set. Definition 1 A mapping T : K → X∗ is said to be monotone if 〈T (x)− T (y), x− y〉 ≥ 0 , ∀ x, y ∈ K .